Change of Base Formula

The change of base formula, as its name suggests, is used to change the base of a logarithm. We might have noticed that a scientific calculator has only "log" and "ln" buttons. Also, we know that "log" stands for a logarithm of base 10 and "ln" stands for a logarithm of base e. But there is no option to calculate the logarithm of a number with any other bases other than 10 and e.

The change of base formula solves this issue of changing base from e to 10, and from base 10 to e. Also, it is used in solving several logarithms problems. Let us learn the change of base formula along with its proof and a few solved examples.

What Is Change of Base Formula?

The change of base formula is used to write a logarithm of a number with a given base as the ratio of two logarithms each with the same base that is different from the base of the original logarithm. This is a property of logarithms. You can see the change of base formula here.

Change of Base Formula

Change of base formula can be represented as follows. Here the base of the given logarithm is also changed to a logarithm with a new base. The basic logarithm with a base is transformed to two logarithms with a new and same base. The change of base formula is:

log\(_b\) a = [log\(_c\) a] / [log\(_c\) b]

In this formula,

• The argument of the logarithm in the numerator is the same as the argument of the original logarithm.
• The argument of the logarithm in the denominator is the same as the base of the original logarithm.
• The bases of both logarithms of numerator and denominator should be the same and this base can be any positive number other than 1.

Note: Another form of this formula is, log\(_b\) a · log\(_c\) b = log\(_c\) a, which is also widely used in solving the problems.

Change of Base Formula Derivation

Change of base formula derivation can be understood from the following steps. Let us assume that

log\(_b\) a = p,log\(_c\) a = q, and log\(_c\) b = r.

By converting each of these into exponential form, we get,

a = b^p, a = c^q, and b = c^r.

From the first two equations,

b^p = c^q

Substituting b = c^r (which is from third equation) here,

(c^r)^p = c^q

c^rp = c^q (Using a property of exponents, (a^m)ⁿ = a^mn)

pr = q

p = q / r

Substituting the values of p, q, and r here,

log\(_b\) a = [log\(_c\) a] / [log\(_c\) b]

Let us see the applications of the change of base formula in the following section.

Examples Using Change of Base Formula

Example 1: Evaluate the value of log\(_{64}\) 8 using the change of base formula.

Solution:

We will apply the change of base formula (by changing the base to 10). Note that log\(_{10}\) is same as log.

log\(_{64}\) 8 = [log 8] / [log 64]

= [log 8] / [log 8²]

= [log 8] / [2 log 8] [∵ log a^m = m log a]

= 1 / 2

Answer: log\(_{64}\) 8 = 1 / 2.

Example 2: Caluclate log\(_9\) 8 using the calculator. Round your answer to 4 decimals.

Solution:

We cannot calculate log\(_9\) 8 directly using the calculator because it doesn't have a button named log\(_9\). Thus, we apply the change of base formula first.

log\(_9\) 8 = [log 8] / [log 9]

=  [0.903089...] / [0.95424...]

≈ 0.9464

Answer: log\(_9\) 8 ≈ 0.9464.

Example 3: Evaluate the value of log\(_3\) 2 · log\(_4\) 3 · log\(_5\) 4.

Solution:

By alternate form of the change of base formula, log\(_b\) a ⋅ log\(_c\) b = log\(_c\) a. We apply this twice to evaluate the given expression.

log\(_3\) 2 · log\(_4\) 3 · log\(_5\) 4

= log\(_4\) 2 · log\(_5\) 4

= log\(_5\) 2

Answer: log\(_3\) 2 · log\(_4\) 3 · log\(_5\) 4 = log\(_5\) 2.